ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Democritus

Updated on June 20, 2010

The beginnings of the integral calculus do not belong with the great 17th century mathematicians, Newton and Leibniz, or with their immediate precursors such as Cavalieri, nor even to ‘The Method’ of Archimedes. They can be traced back to a little known Greek mathematician named Democritus.

Democritus was born in about 460 BC in the town of Abdera, Thrace, in the northern part of Greece. His Abderic school of thought believed in atomism. They believed that everything, even the mind and the soul, is made up of atoms that move around in a void. These atoms are particles that come in various shapes and sizes, but are indivisible. His idea for the physical world is not very different from the modern concept of the atom.

According to Archimedes, Democritus applied the idea of atomism to geometry by using infinitesimals to find the formula for the volume of a cone. He was the first mathematician to give the formula for the volume of a pyramid – not just a square-based pyramid - but one whose base is a regular polygon with any number of sides, including the cone (which is just a circular-based pyramid).

The formula for the volume of a square-based pyramid was known to the Egyptians and Democritus may well have learned of it during his travels. But he expanded on this by giving the formula

Volume = 1/3 × Area of base × Height

for any pyramid. The fact that he inferred the formula for the volume of a cone by regarding its base as a regular polygon with an infinite number of sides of infinitesimal length is the first example of a technique that forms the basis of the integral calculus.

A contribution of Democritus to mathematics

Democritus showed that the volume of each of the first three pyramids is given by the formula

Volume = 1/3 × Area of base × Height

In the case of the pyramid whose base is a regular 20-sided polygon, for obvious reasons, I have not drawn in all the sloping edges from the vertex to each of the 20 points around the base. It is clear, however, that there are 20 straight edges around the base and that the polygon is very close to being a circle. Imagine a similar pyramid whose base is a 100-sided regular polygon – it would be almost indistinguishable from a circle.

Democritus used his atomist ideas to show that, as the number of sides of the polygon increases, the polygon gets closer and closer to being a circle, and becomes a circle when the number of sides is infinite. In modern terms we would say that ‘in the limit, as the number of sides of the polygon tends to infinity, the polygon tends to a circle’.

Of course, in the case of the circular based pyramid – the cone – the formula becomes

Volume = 1/3 × πr2 × h where r is the radius of the base.

Comments

    0 of 8192 characters used
    Post Comment

    • profile image

      Wes 6 years ago

      Great information for my report

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: "https://hubpages.com/privacy-policy#gdpr"

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)