Рассказ про архимеда краткий

Полное имя:	Архимед
Дата рождения:	287 г. до н. э.
Место рождения:	Сиракузы, Италия
Деятельность:	Ученый, математик, физик, астроном, инженер, изобретатель
Дата смерти:	212 г. до н. э. (≈75 лет)

Ранние годы и молодость

Архимед родился в 287 году до н. э. в колонии Сиракузы на греческом острове Сицилия. По неподтвержденной информации отец его, Фидий, был учителем астрономии и математики. Семья, в которой воспитывался Архимед, была бедной. Отец не мог дать сыну достойное образование, которым в то время считались занятия литературой и философией. Он сам его потихоньку обучал тому, что знал. Поэтому и проявилась ранняя тяга Архимеда к математике.

Архимед являлся родственником царя Сиракуз, Гиерона. Благодаря этому будущий ученый смог поехать в главный научный центр Античности — Александрию.

Архимед стал общаться с другими учеными Александрийского мусейона. Здесь он часто приходил в знаменитую библиотеку, который содержал более 700 тысяч рукописей. Архимед много читал, знакомился с трудами ученых. Среди них Демокрит, Евдокс, о которых он много упоминал в своих записях. Ученый до конца жизни дружил с астрономом Кононом, ученым Эратосфеном. Две свои работы он даже начал со слов «посвящается Эратосфену». Это известные его труды «Метод механических теорем» и «Задача о быках».

Смерть Конона в 220 году до н. э. очень огорчила Архимеда. Однако он активно продолжал общение с его учеником Досифеем, и многие сохранившиеся трактаты ученого тех лет начинаются с фразы: «Архимед приветствует Досифея».

Будущий великий ученый стремился на родину и, завершив учебу, вернулся на Сицилию. Царь Сиракуз поощрял своего родственника к занятию наукой и создавал ему все условия для этого. Он хотел, чтобы Архимед применял свои достижения и открытия на практике. Именно он способствовал созданию Архимедом механизмов, работа которых повергла в шок современников и принесла всемирную славу ученому уже при жизни. Благодаря его изобретениям, о нем слагались легенды. Царь Сиракуз очень гордился достижениями Архимеда.

Громко об Архимеде заговорили после истории с короной царя. До Гиерона дошла информация, что золото в короне, которую он приказал сделать для себя, частично заменили серебром. Он разгневался и поручил Архимеду определить, правда ли это. Масса короны соответствовала количеству выделенного на её изготовление чистого золота. Архимед согласился выполнить приказ царя, но перед этим решил помыться ванной. Зайдя в воду, он увидел, как из неё вытекает вода. В эту секунду его осенила идея, и он с криком «Эврика!» выскочил из ванны и побежал к царю. Вот так внезапная мысль Архимеда стала началом развития гидростатики. Таким образом был доказан и обман ювелира.

Еще одна легенда про Архимеда гласит, что он написал Гиерону, что сможет сдвинуть с места любой груз, и что если бы он распоряжался другой землей, на которую можно было бы опереться, он сдвинул бы с места и эту землю. В ответ на вызов Архимеда царь поручил вытащить на берег большой грузовой корабль. Его трюм наполнили грузом и посадили на корму много матросов. Сев неподалеку, Архимед начал тянуть прикреплённый к кораблю канат. Корабль сдвинулся с места и стал медленно перемещаться.

В современном звучании известная фраза Архимеда означает: «Дайте мне точку опоры, и я переверну Землю».

Основные достижения

Главные достижения и идеи Архимеда приходятся на 3 век до н. э., а именно:

1. Основание механики и гидростатики.
2. Обоснование понятия «центр тяжести».
3. Расчет числа пи. Верхний предел (22⁄7).
4. Открытие формулы для вычисления объема и площади поверхности сферы.
5. Введение способа записи очень больших чисел.
6. Применение физических принципов (закон рычага) в решении математических задач.
7. Изобретение высокоточной катапульты.
8. Развитие науки в Греции.
9. Изобретение «архимедова винта».

Греческий ученый Архимед, благодаря которому были сделаны многочисленные открытия, стал одним из лучших умов нашего мира. Его труды пересматривались из века в век после его смерти. Ученым, которые не сомневались в правдивости его открытий, все же приходилось гадать, каким путем он к ним пришел. В особенности их волновали математические открытия Архимеда.

Долгое время математические способности Архимеда не были известны. Лишь в 1906 году профессор Йохан Хейберг обнаружил в турецком городе Константинополе книгу, содержащую семь трактатов под авторством Архимеда, включая «Метод», который считался утерянным на протяжении тысячелетия.

Этот труд положен в основу почти всем математическим новшествам двадцатого века.

К известным трактатам Архимеда, написанным на греческом языке, относятся:

• Измерение круга.
• О коноидах и сфероидах.
• О шаре и цилиндре.
• Квадратура параболы.
• О счислении песчинок.
• О равновесии плоских фигур.
• Метод механических теорем.
• О плавающих телах.

Важным событием является выход в печать работ Архимеда в 1544 году. До этого времени их не печатали.

Архимед провел большую часть жизни в Сиракузах. Он активно защищал город во время осады его римлянами в 213 году до н. э. Позже, когда Сиракузы все же были захвачены римлянами под предводительством полководца Марка Клавдия Марцеллы в 212 году до н. э., Архимед был убит римским солдатом. Он убил Архимеда вопреки приказу своего командования не убивать ученого.

Личная жизнь

Единственный человек, который подробно описал биографию Архимеда – его друг Гераклид. К сожалению, записи были утеряны, в связи с чем до наших дней дошла лишь часть информации из жизни Архимеда. Биография Архимеда известна из трудов Цицерона, Тита, Полибия, Ливия и других авторов, которые жили позже него и, в основном, представлена его достижениями. О личной жизни, семье, детях ничего не известно.

В биографии Архимеда, написанной знаменитым Плутархом, упоминается, что он был настолько увлечен наукой, что ничего вокруг себя не замечал. Он забывал есть, менять одежду и приводить себя в порядок.

Архимед, как сообщается, будучи в молодом возрасте, любил посещать библиотеки. В особенности Александрийскую библиотеку с ее просторными залами, которая стала центром науки для ученых древнего мира. Архимед пополнял свои знания ежедневно и мечтал о новых открытиях в науке.

След в истории

Архимед был, без преувеличения, величайшим ученым. Он был математиком, астрономом, физиком, инженером, изобретателем, а также выдающимся представителем своего времени, благодаря трудам и изобретениям которого мир опередил время.

Вклад Архимеда в науку бесценен. Он сделал для прогресса науки больше, чем большинство учёных до него, и после. Его выдающийся ум принёс беспрецедентную пользу человечеству и заставил говорить о нем все последующие поколения.

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Архимед (287-212 гг. до н. э.) – древнегреческий ученый и инженер. Автор множества открытий в сфере геометрии, предвосхитил многие идеи математического анализа. Сделал множество открытий в области геометрии, предвосхитил многие идеи математического анализа. Заложил основы механики, гидростатики, был автором ряда важных изобретений.

С именем Архимеда связаны многие математические понятия. Наиболее известно приближение числа π, которое называется Архимедовым числом. Кроме того, имя Архимеда носят граф, ещё одно число, копула, аксиома, спираль, тело, закон и другие.

В биографии Архимеда есть много интересных фактов, о которых мы расскажем в данной статье.

Итак, перед вами краткая биография Архимеда.

Биография Архимеда

О биографии Архимеда нам известно из упоминаний Полибия, Цицерона, Плутарха, Витрувия и других древних авторов. Поскольку все они жили гораздо позже его, достоверность их сведений оценить сложно.

В своих трудах биографы Архимеда упоминают его достижения в науках, открытия, изобретения и другие интересные факты из жизни ученого. Известно, что Архимед появился на свет 287 г. до н.э. в эллинской колонии Сиракузы, находящейся на востоке Сицилии.

Считается, что отцом Архимеда был астроном и математик Фидий. По словам Плутарха, изобретатель приходился родственником Гиерону II – главе Сиракуз. Очевидно, что его детские годы прошли в этой колонии, после чего он отправился обучаться наукам в Александрию.

Стоит отметить, что Архимед сам указывал в своих работах, что обучался математике в Александрии. При этом начальное образование он, по-видимому, получил у отца. Спустя какое-то время ученый вернулся в Сиракузы и жил там до самой смерти.

Механика

Архимед проявлял большой интерес к механике, вследствие чего сконструировал немало разных устройств. Он детально описал способ действия рычага, который многократно использовал на практике.

И хотя о «рычаге» было известно задолго до Архимеда, именно ему удалось популяризировать его и продемонстрировать его эффективность в разных областях. В частности, ему удалось сконструировать ряд различных блочно-рычажных механизмов в порту Сиракуз.

Посредством таких приспособлений соотечественники Архимеда смогли быстрее и проще перемещать тяжелые грузы. Также он является автором известного механизма, названного его именем – «Архимедов винт». С помощью данного устройства человек мог в одиночку выкачивать воду.

Такой винт позже начали использовать в самых разных сферах, включая перекачку жидкостей и сыпучих веществ, как, например, уголь и зерно. Важно не забывать и теоретические разработки Архимеда.

Опираясь на все тот же закона рычага, изобретатель издал научную работу «О равновесии плоских фигур». Здесь он обосновал доказательство того, что на равных плечах, равные тела по необходимости уравновесятся. В другом труде – «О плавании тел», он описал подобный принцип.

Математика

На протяжении всей биографии Архимед проявлял огромный интерес к точным наукам. По воспоминаниям Плутарха, когда он начинал работать, то забывал о еде и полностью погружался в вычисления. В сфере математики его больше всего интересовали вопросы математического анализа.

Архимед нашел новые эффективные способы подсчета объемов и площадей. Ему удалось усовершенствовать метод исчерпывания Евдокса Книдского, вследствие чего он мастерски применял его на практике. И хотя еще до Архимеда была сформулирована теория интегрального исчисления, именно его труды легли в основу данной теории.

Одновременно с этим, Архимед заложил базу для дифференциальных вычислений. Он смог определить, что объемы конуса и шара, вписанных в цилиндр, и самого цилиндра имеют соотношение – 1:2:3. До этого еще никому не удавалось вычислить поверхность и объем шара.

Интересен факт, что Архимед завещал выбить на собственном надгробии шар, вписанный в цилиндр. Кроме этого, математик смог узнать площадь поверхности для сегмента шара и витка открытой им «спирали Архимеда».

Отдельного внимания заслуживает вычисленное Архимедом отношение длины окружности к диаметру. В труде «Об измерении круга» он подробно описал свое легендарное приближение для числа «Пи» (π).

Чтобы доказать свои предположения, Архимед построил для круга вписанный и описанный 96-угольники, после чего определил длины их сторон. Параллельно с этим, он научно обосновал, что площадь круга равна числу π, умноженному на квадрат радиуса круга. Именно так появилось знаменитая формула – πr².

Закон Архимеда

Под законом Архимеда подразумевается следующее: на тело, погруженное в жидкость или газ, действует выталкивающая или подъемная сила, равная массе объема жидкости или газа, вытесненного частью тела, погруженной в жидкость или газ.

Согласно известной легенде, Архимед якобы открыл свой закон, когда выполнял просьбу Гиерона. Правитель хотел узнать, не обманывает ли его работник, выполнявший заказ на изготовление золотой короны. Он понимал, что работник мог присвоить себе часть предоставленного золота, а вместо желтого метала подмешать серебро.

Чтобы решить эту непростую задачу, Архимед отлил 2 равноценных по весу слитка из серебра и золота. Затем он поочередно опустил слитки в емкость, до краев заполненную водой, что позволило ему узнать какое количество воды вытеснил каждый из слитков.

Благодаря этому, Архимед вычислил сколько воды вытесняет золото. Потом он просто поместил корону в емкость с водой и узнал, соответствует ли ее вес вытесняемой жидкости. Опасения Гиерона оправдались, ювелир действительно смошенничал, присвоив часть золота.

По легенде, математик сделал свое открытие во время нахождения в ванной. Когда Архимед садился в воду он заметил, что вес его тела поднял ее уровень. С громким возгласом «Эврика», мужчина выскочил из ванны и нагим помчался к правителю.

Астрономия

Архимед стал автором не менее 3-х работ по астрономии. Он пытался узнать размеры Вселенной, и был первым изобретателем планетария. При движении данного устройства можно наблюдать следующие явления:

• восход Луны и Солнца;
• вращение 5 планет;
• исчезновение Луны и Солнца за горизонтом;
• фазы и затмения Луны.

Архимед стремился определить максимально точные расстояния до планет. Современные ученые полагают, что Землю он считал центром мироздания. Изобретатель думал, что Венера, Марс и Меркурий двигаются вокруг Солнца, а уже вся эта система движется вокруг Земли.

Личная жизнь

О личной биографии Архимеда мы практически ничего не знаем. Еще при жизни о нем начали слагать легенды. По одной из них, когда Гиерону не удавалось спустить на воду многопалубное судно «Сиракузия», он решил обратиться за помощью к математику.

Из нескольких блоков Архимед якобы сконструировал механизм, который позволил спустить корабль на воду одним касанием руки. Бытует мнение, что именно тогда он сказал свою популярную фразу: «Дайте мне точку опоры, и я переверну мир».

Смерть

В 212 г. до н.э. в разгар Второй Пунической войны Сиракузы были осаждены римскими войсками. Архимед постоянно задействовал свой инженерный талант, чтобы оказать помощь соотечественникам. К примеру, он соорудил метательные машины, посредством которых римлян забрасывали крупными камнями.

Тогда же Архимед сконструировал огромные краны, с помощью которых греки могли цеплять крючьями вражеские корабли и отбрасывать их обратно в море. Римские судна переворачивались набок и нередко шли на дно.

Когда римляне поняли, что вряд ли смогут взять город штурмом, они избрали тактику осады. Осенью 212 г. до н.э. Сиракузы пали вследствие измены. Во время захвата колонии Архимед был убит. Есть несколько версий смерти гениального ученого.

Согласно одной их них Архимеда убили в его лаборатории. Он якобы был так сильно увлечен работой, что отказался сразу следовать за римским воином, которому надлежало доставить пленника к начальству. В результате, разгневанный неподчинением солдат заколол Архимеда.

Фото Архимеда

Если вам понравилась краткая биография Архимеда – поделитесь ею в соцсетях. Если же вам нравятся биографии известных людей или интересные истории из их жизни, – подписывайтесь на сайт InteresnyeFakty.org.

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Краткая биография Архимеда

Пожалуй, при слове изобретатель или каком-то подобном довольно часто в уме появляется имя Архимеда. Этот древний мыслитель действительно был выдающимся изобретателем и оставил существенное количество открытий, которые повлияли на развитие всего человечества в дальнейшем.

Архимед родился в 287 году до новой эры на территории острова Сицилия в столице – Сиракузы. Он родился в довольно знатной семье, отец его сам был математиком, а также он был известен тирану того города Гиерону Второму. Оба они с ранних лет заметили в мальчике склонность к познаниям и отправили в подростковом возрасте Архимеда учиться в Александрию Египетскую, именно там была крупнейшая библиотека, которую потом сжег Герострат, чтобы прославиться.

После обучения, за период которого он познакомился со множеством ученых мужей своего времени и усвоил передовые идеи, Архимед возвращается на родину и фактически поступает на службу к Гиерону. Тиран всячески хочет, чтобы Архимед начал разрабатывать всяческие военные новшества для острова, а молодой ученый придерживается миролюбивых воззрений и хочет заняться только изучением мира. Итак, Архимед остается на острове и начинает совершать свои открытия, многие из которых оказываются итогом работы с Гиероном, к примеру, именно он хотел чтобы молодой математик определил состав короны, но, не повреждая сам предмет.

Именно тогда и появилось изобретение о вытеснении телами разного объема воды, при идентичной массе. Помимо этого Архимед сделал множество открытий в математике, которые не много ни мало опередили эпоху на пару тысяч лет. Именно так, некоторые идеи, такие как полуправильные многогранники или использование парабол и гипербол для решения уравнений, ученые смогли по достоинству оценить и развить только в новом времени, после средневековья.

В 212 году Сиракузы оказались под натиском римских войск. Тогда шла вторая Пуническая война и Сицилия была в невыгодном положении между империей и Карфагеном. Архимед сделал немало военных изобретений для того чтобы отстоять собственный город (метательные орудия, отражающие медные пластины и многое другое) тем не менее Сиракузы пали, а Архимед погиб от руки римского солдата.

Биография 2

Точная биография Архимеда, к сожалению, неизвестна. Учёными и археологами разных эпох были приведены разные факты из его жизни, но и они основаны на трудах людей, живших много позднее Архимеда. Согласно самой распространённой версии, будущий математик родился в 287 году до н.э. Местом рождения были Сиракузы (остров Сицилия). Отец мальчика, астроном и математик, отправил сына обучаться в Александрию. Излюбленным местом будущего физика и математика стала библиотека Александрии, где он изучал труды и сочинения Демокрита, Евдокса и многих других учёных. Там же Архимед заводит знакомства, которые пронесёт через всю жизнь.

Молодой человек с юности любил математику. Всё время он посвящал разработкам в области арифметики, алгебры и геометрии. Специалисты этих областей смогли понять, классифицировать и развить его идеи только к 17 веку. Архимед решал сложнейшие уравнения, находя решения графически. Вычислял площади, объёмы разного рода геометрических фигур. Собирал и обобщал в единые принципы и формулы  уже известные методы вычисления. Выводил и доказывал постулаты и аксиомы, которые мало того, что не опровергнуты, но и взяты за основу современными учёными. Одно из важнейших его достижений в геометрии, по его же словам, заключалось в нахождении площади поверхности и объёма сферы. Также он вывел формулы расчетов объёмов параболоида, гиперболоида вращения и эллипсоида. До Архимеда этих вычислений не совершал ни один математик.

Помимо арифметики, алгебры и столь любимой им геометрии, Архимед применял свои знания в области механики и физики, изобретая и совершенствуя уже существующие конструкции и механизмы. К примеру, известный до его рождения рычаг Архимед усовершенствовал, рассчитав его возможности и применив на практике в порту Сиракуз. Некоторые приспособления и механизмы, основанные на принципе рычага, с тех пор существенно облегчали тяжёлую работу.

Астрономия также не оставила его равнодушным. Учёный занимался определением расстояния между космическими объектами, хотя и делал это с ошибочной точки зрения. Ведь в 3 веке до н.э. была распространена геоцентрическая теория существования мира. Впрочем, позднее Архимед преподнёс в одном из своих трудов гелиоцентрическую теорию.

В его честь названы цепь гор и кратер на поверхности Луны, астероид, улицы в нескольких городах России и она улица в Амстердаме. Погиб Архимед во время военных действий при наступлении римлян на Сиракузы. Для победы своей Родины учёный создал метательные механизмы. Римские войска существенно пострадали от этих машин. Решено было держать город в осаде. В 212 году до н.э. Сиракузы сдались, а Архимед был убит.

Биография по датам и интересные факты. Самое главное.

Другие биографии:

• Карл Брюллов

  Брюллов родился в Петербурге в 1799 году и оставил мир неподалеку от города Лацио и Рима в коммуне Манциана в 1852 году. Он был третьим сыном в семье преподавателя Академии художеств

• Александр Федорович Керенский

  Керенский родился не в самой богатой семье, но и не в очень бедной, 1881 года, в мае, в городе Симбирск. Кроме того, в этом городе также родился Ленин. Родители Александра хорошо дружили с родителями Ленина.

• Винсент Ван Гог

  Ван Гог родился в 1853 году, а умер в 1890. Он вдохновлялся такими велики художниками, как Миле и Сардо и ориентироваться на них в своём творчестве. Как художник Ван Гог начал с зарисовки различных сцен из жизни

• Василий Белов

  Василий Иванович Белов — писатель, сценарист, публицист и прозаик. Появился на свет 23 октября в 1932 году в селе Тимониха Волгоградской области в Российской империи.

• Александр III

  Александр III – Всероссийский император, правивший страной в период с 1 марта 1881 по 20 октября 1894 года. Известен как Царь-Миротворец, так как за все время его правления страна не знала сражений и войн.

Краткая биография Архимеда самое главное

Знаменитый учёный эпохи Античности, Архимед родился в 287 году до н.э. на Сицилии, в городе Сиракузы. Об этом рассказал миру в своих трудах Иоанн Цеце. Сам Архимед не оставил потомкам автобиографию, воспоминания, мемуары. Он полностью посвятил свою жизнь математике, физике, геометрии, инженерии. Талантом писателя пользовался, но для других целей.

По площади Сиракузы были самым крупным населённым пунктом античного мира. Самыми крупными культурными центрами были, конечно же, Афины, Рим. Но и в Сиракузах цивилизация процветала: у будущего учёного в детстве была возможность получить хорошее образование, много времени уделять урокам, притом предметам в чём-то абстрактным, философским. Ораторское искусство, популярное в те времена, не увлекло его. Скорее всего, потому что Сиракузы находились вдали от столицы. Он не стал великим архитектором, не разбогател за счёт торговых сделок, земледелия. Восхищённый миром чисел и идей, его призрачностью и реалистичностью, он изобретал, открывал невидимые законы.

Большинство исследователей считают, что его отец – математик, астроном Фидий. Сам Архимед упоминает его в трактате «Исчисление песчинок». Больше нигде, из найденных документов той эпохи, это имя не звучит. Но понять, о ком именно идёт речь сложно – текст можно перевести по-разному. В одной интерпретации в нём о Фидии написано, что тот был сыном Акупатра, а в другой – отцом Архимеда. Плутарх утверждал, что Архимед был родом из весьма влиятельной семьи: правитель Сиракуз, Гиерон II был его родственником.

В юности будущий великий учёный отправился в Египет, город Александрию, чтобы начать свою карьеру в области науки. Он многого не знал, было чему поучиться. Здесь располагалась великолепная библиотека, с огромным количеством рукописей. В этом городе также проживали многие известные учёные. В Александрии Архимед подружился с небезызвестным астрономом Кононом Самосским, поэтом, математиком, географом Эратосфеном Киренским.

После обучения он снова приехал в Сиракузы. Очень быстро создал себе имя, репутацию гения. Успех ему принесли, однако, не рассуждения. Его открытия были полезными государству. Он, не стесняясь, применял их на практике. Например, считается, что он открыл закон Архимеда, измеряя корону короля, пытаясь выяснить, сколько в ней золота, а сколько примесей. Впрочем, вёл он и переписку с другими учёными, обсуждал достижения, делился своими успехами.

В те времена постоянно шла война между разными государствами за территорию. Но это не мешало цивилизации развиваться. Древняя Греция была серьёзным противником. В ней жили славные воины, трудолюбивые земледельцы. Архимеду же была свойственна фанатичность, он без ума был влюблён в науку, служил истине. В 212 году до н.э. началась война за Сиракузы с римлянами. Именно Архимед придумал грозное оружие, метательные машины с прицельным огнём. Благодаря его усилиям атаки были отражены. Но город взяли в осаду, и вскоре, всё в том же 212 году, он был захвачен хитростью – помог предатель. Архимед погиб в той битве. Ему было 75 лет. О сражении за Сиракузы, изобретениях учёного рассказано в трудах Плутарха, Тита Ливия, Анфимия Траллийского.

Самое важное. Для детей и школьников и его открытие

Биография Архимеда о главном

 Архимед является выдающимся древнегреческим математиком, инженером и изобретаем своего времени. Родился он в 287 году до н.э. в городе Сиракузы на Сицилии. Отец Фидий был физиком и математиком, находящийся при дворе.

Архимед получил превосходное образование, но он понимал, что всё-таки ему недостаёт теоретических знаний, которые были даны ранее. Поэтому проводил в Александрийской библиотеке всё своё свободное время. После окончания учёбы стал работать при дворе – астрономом, создал планетарий. Так как в то время было модно изучать астрономию, то все считали, что Солнце и Луна вращаются вокруг Земли, но только Архимед предположил, что именно все планеты кружатся вокруг Солнца. Помимо этого он изучал механику, физику и математику свои труды он изложил в своих работах «О равновесии плоских фигур», далее последовало сочинение «Об изменении круга».

У ученого много открытий, который он посветил своей родине. Ему удалось воссоздать целую рычагово-блочных механизмом, которые позволяют перевозить тяжелые грузы намного быстрее.

Так же у Архимеда существуют множество работ связанных с алгеброй, геометрией, арифметикой. Разработал всесторонний  метод вычисления площадей разнообразных фигур. Создал теорию об уравновешивании равных тел.

Одной из версий смерти является, что убили великого ученого в период Второй Пунической войны, когда после продолжительной обороны взяли Сиракузы. Марк Клавдий Марцелл отдал приказ  разыскать Архимеда и привести к нему. За ним послали солдата, он направился в дом ученого, который в этот период производил математические расчёты. Солдат потребовал явиться к римлянам, на что Архимед ответил, только после того как закончу расчёты, но разозлившись солдат зарезал мечом самого гениального человека такого времени. Он прожил всего лишь 75 лет.

Самое важное.  Для детей и школьников и его открытие

Интересные факты и даты из жизни

Архимед — доклад сообщение

Архимед – ученый родом из Сиракуз, который прожил свой век, начиная с 287 до 212 года до новой эры. Еще мальчиком он проявлял способности к познанию мира и царь Сиракуз отправил его «на большую землю» учиться к лучшим умам того времени, в частности с 272 года Архимед учится у Эвклида и пользуется Александрийской библиотекой.

В 20 лет молодому ученому требуется возвращаться обратно на родину, хотя Эвклид и отговаривал его от служения царю, ведь он станет использовать его знания для того чтобы упрочить собственную власть, а не для того чтобы упрочить и распространить знания.

По дороге обратно корабль Архимеда терпит крушение, и он чудом спасается. После чего он изобретает так называемый архимедов винт – полую трубу с винтом, благодаря которой возможно откачивать существенные объемы воды из тонущего корабля, этот винт потом использовали не только для кораблей, но и в оросительных системах.

Далее царь Сиракуз дал ученому задание: узнать из какого металла сделана его корона. Он сказал не разламывать корону и не повреждать изделие, но узнать, из чего сделан предмет. Тогда и родилась известная легенда с криком «Эврика» (в переводе «нашел») и пониманием как измерить состав изделия через объем вытесняемой воды.

Архимед помимо этого занимался и астрономическими исследованиями, которые были довольно современными для его времени. Он измерял дистанции от Солнца до Земли, придерживался мнения о наличии солнечной системы, которое расходилось с распространенным мнением о Земле, вокруг которой вращаются остальные планеты.

Царь Геирон предоставил ему возможность выполнять изыскания в области математики и других областей знаний. Архимед стал великим ученым и сделал открытия, которые стали базой для математических изысканий следующих времен. Он является создателем ряда фундаментальных математических теорем.

Тем не менее, со временем ученому пришлось отказаться от мирной деятельности. Сиракузы были довольно богатым островом и в Пунические войны выступали против Карфагена, и во второй войне Карфаген во главе с Ганнибалом направил свои корабли к Сиракузам. Поэтому Архимеду предстояло создать множество оборонительных сооружений, которые смогли бы сохранить неприкосновенность острова.

В итоге Архимед создал множество механизмов, но произошел переворот, сына Геирона предали и в итоге город стал союзником Карфагена и против него выступил Рим. Войска империи продолжали атаковать город, хотя Сиракузы отлично оборонялись благодаря изобретениям Архимеда. В итоге житель Сиракуз предал свой город и впустил римские войска, а Архимед погиб от руки римских солдат.

Вариант 2

Многие ученые древности не были поняты своими современниками, так как они родились слишком рано. Чтобы общество могло понять их труды и должным образом оценить их, этим людям следовало бы появиться на свет на много столетий позже. Самым ярким, самым выдающимся ученым древности по праву считается Архимед из Древней Греции.

Он родился в Сиракузах, дату можно назвать лишь примерно – 287 год до нашей эры. Мальчик получил хорошее образование, но все же решил поехать в Александрию, которая была тогда крупным культурным и образовательным центром. Архимед изучал там разные науки, однако больше всего времени он посвятил математике. После окончания учебы он вернулся в Сиракузы.

В родном городе Архимед получил хорошую должность, он стал придворным астрономом. Имея достаточно свободного времени, занялся научной работой. Этот человек был гениален во многих областях, и свидетельством этого являются труды, которые дошли до нас.

Ученому было уже более 70 лет, когда он был назначен ответственным за оборону Сиракуз, так как город пытались захватить римляне. И Архимед выполнил эту задачу блестяще, очередной раз подтвердив свою гениальность. Он изобрел мощную метательную машину и создал несколько образцов. Более трех лет римляне не могли захватить город, а когда это случилось, то первых солдат, вступивших в Сиракузы, встретил град камней из метательных машин Архимеда.

Так что великий ученый был не только теоретиком, но и практиком, инженером. Он использовал рычаги для перемещения тяжестей в порту, во время войны по его приказанию греки поджигали корабли римлян с помощью зеркал, все его изобретения были полезными и нужными, а некоторыми мы пользуемся до сих пор.

Научные труды Архимеда на много веков опередили свое время, например, его работы по математике стали понятны ученым лишь в 17 веке, тогда они и получили свое развитие и продолжение.  Математика была любимой наукой Архимеда. Он заложил основу интегрального исчисления и математического анализа, с помощью его формулы мы считаем радиус окружности, площадь и объем разных фигур. Он изобрел немало механизмов, построил планетарий. Архимед был первым во многих науках. И сейчас ученые используют закон Архимеда, правило Архимеда, теорему Архимеда.

К сожалению, не все труды ученого дошли до наших времен. Но и то, что должно, бесценно для науки.

Умер Архимед в 212 году до нашей эры. Его убили римляне, против которых он так долго воевал. Но имя его помнят до сих пор, как имя одного из самых выдающихся и гениальных людей за всю историю.

3, 5, 6, 7 класс по физике, математике.

Архимед

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Archimedes of Syracuse
Ἀρχιμήδης
Archimedes Thoughtful by Domenico Fetti (1620)
Born	c. 287 BC Syracuse, Sicily
Died	c. 212 BC (aged approximately 75) Syracuse, Sicily
Known for	List Archimedes’ principle Archimedes’ screw Center of gravity Statics Hydrostatics Law of the lever Indivisibles Neuseis constructions[1] List of other things named after him
Scientific career
Fields	Mathematics Physics Astronomy Mechanics Engineering
Influences	Eudoxus
Influenced	Apollonius[2] Hero Pappus Eutocius

Archimedes of Syracuse (;^[3][a] c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.^[4] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,^[5] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.^[6][7] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.^[8][9]

Archimedes’ other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes’ achievements in this area include a proof of the law of the lever,^[10] the widespread use of the concept of center of gravity,^[11] and the enunciation of the law of buoyancy or Archimedes’ principle.^[12] He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

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 Archimedes’ tomb, which was surmounted by a sphere and a cylinder that Archimedes requested be placed there to represent his mathematical discoveries.

Unlike his inventions, Archimedes’ mathematical writings were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes by Eutocius in the 6th century 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 and again in the 17th century,^[13][14] while the discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.^{[15][16][17][18]}

Biography

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in 212 BC.^[9] In the Sand-Reckoner, Archimedes gives his father’s name as Phidias, an astronomer about whom nothing else is known.^[20] A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth.^[21] From his surviving written works, it is clear that he maintained collegiate relations with scholars based there, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene.^[b]

The standard versions of Archimedes’ life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in The Histories by Polybius (c. 200–118 BC), written about 70 years after his death. 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 from the Romans.^[22] Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse’s defenses, and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could be swung around in an arc, and other stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes’ inventiveness.^[23]

Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero found what was presumed to be Archimedes’ 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. The tomb carried a sculpture illustrating Archimedes’ favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of the cylinder including its bases.^[24][25] He also mentions that Marcellus brought to Rome two planetariums Archimedes built.^[26] The Roman historian Livy (59 BC–17 AD) retells Polybius’ story of the capture of Syracuse and Archimedes’ role in it.^[22]

Plutarch (45–119 AD) wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse.^[27] He also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Marcellus was reportedly angered by Archimedes’ death, as he considered him a valuable scientific asset (he called Archimedes «a geometrical Briareus») and had ordered that he should not be harmed.^[28][29]

The last words attributed to Archimedes are «Do not disturb my circles» (Latin, «Noli turbare circulos meos«; Katharevousa Greek, «μὴ μου τοὺς κύκλους τάραττε»), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch’s account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in Memorable Doings and Sayings, «… sed protecto manibus puluere ‘noli’ inquit, ‘obsecro, istum disturbare’» («… but protecting the dust with his hands, said ‘I beg of you, do not disturb this‘«).^[22]

Discoveries and inventions

Archimedes’ principle

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 votive crown for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be used; Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith.^[30] 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.

In Vitruvius’ account, Archimedes noticed while taking a bath 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 crown’s volume. For practical purposes water is incompressible,^[31] so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass 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: «εὕρηκα, heúrēka!, lit. ‘I have found [it]!’).^[30] The test on the crown was conducted successfully, proving that silver had indeed been mixed in.^[32]

The story of the golden crown does not appear anywhere in Archimedes’ known works. The practicality of the method it describes has been called into question due to the extreme accuracy that would be required while measuring the water displacement.^[33] 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.^[34] Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing the crown on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.^[12] Galileo Galilei, who in 1586 invented a hydrostatic balance for weighing metals in air and water inspired by the work of Archimedes, 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.»^[35][36]

Archimedes’ screw

A large part of Archimedes’ work in engineering probably arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hiero 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.^[37] 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, 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. Archimedes’ screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described in Roman times by Vitruvius, Archimedes’ screw may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.^[38][39] The world’s first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.^[40]

Archimedes’ claw

Archimedes is said to have designed a claw as a weapon 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.^[41]

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.^[42]

Heat ray

Archimedes may have written a work on mirrors entitled Catoptrica,^[c] and later authors believed he might have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. Lucian wrote, in the second century AD, that during the siege of Syracuse Archimedes destroyed enemy ships with fire. Almost four hundred years later, Anthemius of Tralles mentions, somewhat hesitantly, that Archimedes could have used burning-glasses as a weapon.^[43] The presumed device, often called the «Archimedes heat ray«, focused sunlight onto approaching ships, causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace.^[44]

Archimedes’ purported heat ray has been the subject of an 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, mostly with negative results.^[45][46] 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, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire.^[47]

Lever

While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes.^[48] Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas.^[49][50] There are several, often conflicting, reports regarding Archimedes’ feats using the lever to lift very heavy objects. 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.^[51] According to Pappus of Alexandria, Archimedes’ work on levers caused him to remark: «Give me a place to stand on, and I will move the Earth» (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω).^[52] Olympiodorus later attributed the same boast to Archimedes’ invention of the baroulkos, a kind of windlass, rather than the lever.^[53]

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.^[54]

Astronomical instruments

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus’ heliocentric model of the universe, in the Sand-Reckoner. Without the use of either trigonometry or a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves),^[55][56] applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error.^[20] 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.^[21]

Cicero’s De re publica portrays a fictional conversation taking place in 129 BC, after the Second Punic War. General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms after capturing Syracuse in 212 BC, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also 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:^[57][58]

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.

This is a description of a small planetarium. Pappus of Alexandria reports on a treatise by Archimedes (now lost) dealing with the construction of these mechanisms entitled On Sphere-Making.^[26][59] Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose.^[60] Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing.^[61] 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.^[62][63]

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes «placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life»,^[28] though some scholars believe this may be a mischaracterization.^[64][65][66]

Method of exhaustion

Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus.^[6] 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 areas of figures and the value of π.

In Measurement of a Circle, he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.^[67] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ().

Archimedean property

In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.^[68]

Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained 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.»^[69] It is possible that he used an iterative procedure to calculate these values.^[70][71]

The infinite series

In 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:

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 whose third vertex is where the line that is parallel to the parabola’s axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

Myriad of myriads

In The Sand Reckoner, Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. 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 itself derives from the Greek μυριάς, murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×10⁶³.^[72]

Writings

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.^[73] Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors.^[9] 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.^[c]

Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), 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 Thābit ibn Qurra (836–901 AD), and into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).^[74][75]

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.^[76]

Surviving works

The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).^[77][78]

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 (π), showing that it is greater than 223/71 and less than 22/7.

The Sand Reckoner

In this treatise, also known 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. 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 is the only surviving work in which Archimedes discusses his views on astronomy.^[79]

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, which states that:

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.^[80]

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 in one of his proofs by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

On the Sphere and Cylinder

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 4/3πr³ 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.

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 modern 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 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

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. 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. Archimedes’ principle of buoyancy is given in this work, stated as follows:

Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in sense 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.

Ostomachion

Also known as Loculus of Archimedes or Archimedes’ Box,^[81] 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.^[82] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.^[83] 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).^[84] Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον, ‘bone’) and machē (μάχη, ‘fight’).^[81]

The cattle problem

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. 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. A. Amthor first solved this version of the problem^[85] in 1880, and the answer is a very large number, approximately 7.760271×10²⁰⁶⁵⁴⁴.^[86]

The Method of Mechanical Theorems

This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses indivisibles,^[6][7] and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He 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 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. 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.^[87]

It has also been claimed that the formula for calculating the area of a triangle from the length of its sides was known to Archimedes,^[d] though its first appearance is in the work of Heron of Alexandria in the 1st century AD.^[88] 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.^[89][90]

Archimedes Palimpsest

The foremost document containing Archimedes’ work is the Archimedes Palimpsest. 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 by Papadopoulos-Kerameus.^[91][92] 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.^[91][93] 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 $2 million.^[94]

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 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.^[95] It has since returned to its anonymous owner.^[96][97]

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

Legacy

Sometimes called the father of mathematics and mathematical physics, Archimedes had a wide influence on mathematics and science.^[98]

Mathematics and physics

Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote:

Any list of the three “greatest” mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and Gauss. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first.^[99]

Likewise, Alfred North Whitehead and George F. Simmons said of Archimedes:

… in the year 1500 Europe knew less than Archimedes who died in the year 212 BC …^[100]

If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself.^[101]

Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes notes:

And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.^[102]

Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.^{[103][104][105]} Galileo called him «superhuman» and «my master»,^[106][107] while Huygens said, «I think Archimedes is comparable to no one» and modeled his work after him.^[108] Leibniz said, «He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.»^[109] Gauss’s heroes were Archimedes and Newton,^[110] 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.»^[111]

The inventor Nikola Tesla praised him, saying:

Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work.^[112]

Honors and commemorations

There is a crater on the Moon named Archimedes (29°42′N 4°00′W / 29.7°N 4.0°W) in his honor, as well as a lunar mountain range, the Montes Archimedes (25°18′N 4°36′W / 25.3°N 4.6°W).^[113]

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri («Rise above oneself and grasp the world»).^{[114][115][116]}

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).^[117]

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.^[118]

See also

Concepts

• Arbelos
• Archimedean point
• Archimedes’ axiom
• Archimedes number
• Archimedes paradox
• Archimedean solid
• Archimedes’ twin circles
• Methods of computing square roots
• Salinon
• Steam cannon
• Trammel of Archimedes

People

• Diocles
• Pseudo-Archimedes
• Zhang Heng

References

Notes

Citations

Further reading

• Boyer, Carl Benjamin. 1991. A History of Mathematics. New York: Wiley. ISBN 978-0-471-54397-8.
• Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
• Dijksterhuis, Eduard J. [1938] 1987. Archimedes, translated. Princeton: Princeton University Press. ISBN 978-0-691-08421-3.
• Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. ISBN 978-0-7660-2502-8.
• Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1-4042-0774-5.
• Heath, Thomas L. 1897. Works of Archimedes. Dover Publications. ISBN 978-0-486-42084-4. Complete works of Archimedes in English.
• Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. ISBN 978-0-297-64547-4.
• Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5.
• Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. ISBN 978-0-7201-2284-8.
• Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 978-0-88385-718-2.

External links

• Heiberg’s Edition of Archimedes. Texts in Classical Greek, with some in English.
• Archimedes on In Our Time at the BBC
• Works by Archimedes at Project Gutenberg
• Works by or about Archimedes at Internet Archive
• Archimedes at the Indiana Philosophy Ontology Project
• Archimedes at PhilPapers
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• «Archimedes and the Square Root of 3». MathPages.com.
• «Archimedes on Spheres and Cylinders». MathPages.com.
• Testing the Archimedes steam cannon Archived 29 March 2010 at the Wayback Machine

Archimedes of Syracuse
Ἀρχιμήδης
Archimedes Thoughtful by Domenico Fetti (1620)
Born	c. 287 BC Syracuse, Sicily
Died	c. 212 BC (aged approximately 75) Syracuse, Sicily
Known for	List Archimedes’ principle Archimedes’ screw Center of gravity Statics Hydrostatics Law of the lever Indivisibles Neuseis constructions[1] List of other things named after him
Scientific career
Fields	Mathematics Physics Astronomy Mechanics Engineering
Influences	Eudoxus
Influenced	Apollonius[2] Hero Pappus Eutocius

Archimedes of Syracuse (;^[3][a] c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.^[4] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,^[5] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.^[6][7] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.^[8][9]

Archimedes’ other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes’ achievements in this area include a proof of the law of the lever,^[10] the widespread use of the concept of center of gravity,^[11] and the enunciation of the law of buoyancy or Archimedes’ principle.^[12] He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

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 Archimedes’ tomb, which was surmounted by a sphere and a cylinder that Archimedes requested be placed there to represent his mathematical discoveries.

Unlike his inventions, Archimedes’ mathematical writings were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes by Eutocius in the 6th century 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 and again in the 17th century,^[13][14] while the discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.^{[15][16][17][18]}

Biography

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in 212 BC.^[9] In the Sand-Reckoner, Archimedes gives his father’s name as Phidias, an astronomer about whom nothing else is known.^[20] A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth.^[21] From his surviving written works, it is clear that he maintained collegiate relations with scholars based there, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene.^[b]

The standard versions of Archimedes’ life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in The Histories by Polybius (c. 200–118 BC), written about 70 years after his death. 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 from the Romans.^[22] Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse’s defenses, and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could be swung around in an arc, and other stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes’ inventiveness.^[23]

Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero found what was presumed to be Archimedes’ 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. The tomb carried a sculpture illustrating Archimedes’ favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of the cylinder including its bases.^[24][25] He also mentions that Marcellus brought to Rome two planetariums Archimedes built.^[26] The Roman historian Livy (59 BC–17 AD) retells Polybius’ story of the capture of Syracuse and Archimedes’ role in it.^[22]

Plutarch (45–119 AD) wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse.^[27] He also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Marcellus was reportedly angered by Archimedes’ death, as he considered him a valuable scientific asset (he called Archimedes «a geometrical Briareus») and had ordered that he should not be harmed.^[28][29]

The last words attributed to Archimedes are «Do not disturb my circles» (Latin, «Noli turbare circulos meos«; Katharevousa Greek, «μὴ μου τοὺς κύκλους τάραττε»), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch’s account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in Memorable Doings and Sayings, «… sed protecto manibus puluere ‘noli’ inquit, ‘obsecro, istum disturbare’» («… but protecting the dust with his hands, said ‘I beg of you, do not disturb this‘«).^[22]

Discoveries and inventions

Archimedes’ principle

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 votive crown for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be used; Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith.^[30] 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.

In Vitruvius’ account, Archimedes noticed while taking a bath 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 crown’s volume. For practical purposes water is incompressible,^[31] so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass 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: «εὕρηκα, heúrēka!, lit. ‘I have found [it]!’).^[30] The test on the crown was conducted successfully, proving that silver had indeed been mixed in.^[32]

The story of the golden crown does not appear anywhere in Archimedes’ known works. The practicality of the method it describes has been called into question due to the extreme accuracy that would be required while measuring the water displacement.^[33] 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.^[34] Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing the crown on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.^[12] Galileo Galilei, who in 1586 invented a hydrostatic balance for weighing metals in air and water inspired by the work of Archimedes, 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.»^[35][36]

Archimedes’ screw

A large part of Archimedes’ work in engineering probably arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hiero 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.^[37] 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, 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. Archimedes’ screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described in Roman times by Vitruvius, Archimedes’ screw may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.^[38][39] The world’s first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.^[40]

Archimedes’ claw

Archimedes is said to have designed a claw as a weapon 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.^[41]

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.^[42]

Heat ray

Archimedes may have written a work on mirrors entitled Catoptrica,^[c] and later authors believed he might have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. Lucian wrote, in the second century AD, that during the siege of Syracuse Archimedes destroyed enemy ships with fire. Almost four hundred years later, Anthemius of Tralles mentions, somewhat hesitantly, that Archimedes could have used burning-glasses as a weapon.^[43] The presumed device, often called the «Archimedes heat ray«, focused sunlight onto approaching ships, causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace.^[44]

Archimedes’ purported heat ray has been the subject of an 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, mostly with negative results.^[45][46] 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, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire.^[47]

Lever

While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes.^[48] Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas.^[49][50] There are several, often conflicting, reports regarding Archimedes’ feats using the lever to lift very heavy objects. 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.^[51] According to Pappus of Alexandria, Archimedes’ work on levers caused him to remark: «Give me a place to stand on, and I will move the Earth» (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω).^[52] Olympiodorus later attributed the same boast to Archimedes’ invention of the baroulkos, a kind of windlass, rather than the lever.^[53]

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.^[54]

Astronomical instruments

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus’ heliocentric model of the universe, in the Sand-Reckoner. Without the use of either trigonometry or a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves),^[55][56] applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error.^[20] 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.^[21]

Cicero’s De re publica portrays a fictional conversation taking place in 129 BC, after the Second Punic War. General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms after capturing Syracuse in 212 BC, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also 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:^[57][58]

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.

This is a description of a small planetarium. Pappus of Alexandria reports on a treatise by Archimedes (now lost) dealing with the construction of these mechanisms entitled On Sphere-Making.^[26][59] Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose.^[60] Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing.^[61] 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.^[62][63]

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes «placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life»,^[28] though some scholars believe this may be a mischaracterization.^[64][65][66]

Method of exhaustion

Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus.^[6] 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 areas of figures and the value of π.

In Measurement of a Circle, he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.^[67] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ().

Archimedean property

In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.^[68]

Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained 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.»^[69] It is possible that he used an iterative procedure to calculate these values.^[70][71]

The infinite series

In 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:

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 whose third vertex is where the line that is parallel to the parabola’s axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

Myriad of myriads

In The Sand Reckoner, Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. 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 itself derives from the Greek μυριάς, murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×10⁶³.^[72]

Writings

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.^[73] Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors.^[9] 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.^[c]

Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), 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 Thābit ibn Qurra (836–901 AD), and into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).^[74][75]

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.^[76]

Surviving works

The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).^[77][78]

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 (π), showing that it is greater than 223/71 and less than 22/7.

The Sand Reckoner

In this treatise, also known 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. 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 is the only surviving work in which Archimedes discusses his views on astronomy.^[79]

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, which states that:

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.^[80]

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 in one of his proofs by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

On the Sphere and Cylinder

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 4/3πr³ 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.

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 modern 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 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

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. 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. Archimedes’ principle of buoyancy is given in this work, stated as follows:

Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in sense 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.

Ostomachion

Also known as Loculus of Archimedes or Archimedes’ Box,^[81] 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.^[82] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.^[83] 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).^[84] Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον, ‘bone’) and machē (μάχη, ‘fight’).^[81]

The cattle problem

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. 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. A. Amthor first solved this version of the problem^[85] in 1880, and the answer is a very large number, approximately 7.760271×10²⁰⁶⁵⁴⁴.^[86]

The Method of Mechanical Theorems

This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses indivisibles,^[6][7] and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He 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 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. 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.^[87]

It has also been claimed that the formula for calculating the area of a triangle from the length of its sides was known to Archimedes,^[d] though its first appearance is in the work of Heron of Alexandria in the 1st century AD.^[88] 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.^[89][90]

Archimedes Palimpsest

The foremost document containing Archimedes’ work is the Archimedes Palimpsest. 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 by Papadopoulos-Kerameus.^[91][92] 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.^[91][93] 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 $2 million.^[94]

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 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.^[95] It has since returned to its anonymous owner.^[96][97]

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

Legacy

Sometimes called the father of mathematics and mathematical physics, Archimedes had a wide influence on mathematics and science.^[98]

Mathematics and physics

Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote:

Any list of the three “greatest” mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and Gauss. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first.^[99]

Likewise, Alfred North Whitehead and George F. Simmons said of Archimedes:

… in the year 1500 Europe knew less than Archimedes who died in the year 212 BC …^[100]

If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself.^[101]

Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes notes:

And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.^[102]

Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.^{[103][104][105]} Galileo called him «superhuman» and «my master»,^[106][107] while Huygens said, «I think Archimedes is comparable to no one» and modeled his work after him.^[108] Leibniz said, «He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.»^[109] Gauss’s heroes were Archimedes and Newton,^[110] 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.»^[111]

The inventor Nikola Tesla praised him, saying:

Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work.^[112]

Honors and commemorations

There is a crater on the Moon named Archimedes (29°42′N 4°00′W / 29.7°N 4.0°W) in his honor, as well as a lunar mountain range, the Montes Archimedes (25°18′N 4°36′W / 25.3°N 4.6°W).^[113]

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri («Rise above oneself and grasp the world»).^{[114][115][116]}

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).^[117]

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.^[118]

See also

Concepts

• Arbelos
• Archimedean point
• Archimedes’ axiom
• Archimedes number
• Archimedes paradox
• Archimedean solid
• Archimedes’ twin circles
• Methods of computing square roots
• Salinon
• Steam cannon
• Trammel of Archimedes

People

• Diocles
• Pseudo-Archimedes
• Zhang Heng

References

Notes

Citations

Further reading

• Boyer, Carl Benjamin. 1991. A History of Mathematics. New York: Wiley. ISBN 978-0-471-54397-8.
• Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
• Dijksterhuis, Eduard J. [1938] 1987. Archimedes, translated. Princeton: Princeton University Press. ISBN 978-0-691-08421-3.
• Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. ISBN 978-0-7660-2502-8.
• Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1-4042-0774-5.
• Heath, Thomas L. 1897. Works of Archimedes. Dover Publications. ISBN 978-0-486-42084-4. Complete works of Archimedes in English.
• Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. ISBN 978-0-297-64547-4.
• Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5.
• Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. ISBN 978-0-7201-2284-8.
• Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 978-0-88385-718-2.

External links

• Heiberg’s Edition of Archimedes. Texts in Classical Greek, with some in English.
• Archimedes on In Our Time at the BBC
• Works by Archimedes at Project Gutenberg
• Works by or about Archimedes at Internet Archive
• Archimedes at the Indiana Philosophy Ontology Project
• Archimedes at PhilPapers
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• «Archimedes and the Square Root of 3». MathPages.com.
• «Archimedes on Spheres and Cylinders». MathPages.com.
• Testing the Archimedes steam cannon Archived 29 March 2010 at the Wayback Machine