ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Claudius Ptolemy

Updated on June 20, 2010

Claudius Ptolemy’s dates are given tentatively as 85 to 165 AD, but we do not know exactly when he was born or died. We do know that he made astronomical observations at Alexandria between 127 and 151 AD, and for that reason he is also often referred to as Ptolemy of Alexandria.

Ptolemy was an astronomer of some note. As a prerequisite to his work on astronomy and the accurate measurements he needed to make, Ptolemy studied trigonometry, and his greatest work was a thirteen book treatise on trigonometry named the Mathematical Syntaxis . This work is more frequently known by the Arabic name Almagest which means ‘the greater’. This name comes about from a comparison with a similar but inferior work written by Aristarchus and others which was referred to as ‘the lesser’.

Ptolemy’s Almagest contains very accurate trigonometric tables. Central to the compilation of these tables was the theorem that has become known as Ptolemy’s Theorem (see below).

Ptolemy also made a contribution to the way we measure angles. The division of the circumference of a circle into 360 equal parts seems to have been in use since the time of Hipparchus around 430 BC, and this idea translated into dividing a complete turn into 360°. However, it was Ptolemy who subdivided the degree into sixty equal parts, named ‘minutes’, and the minute into sixty equal parts named ‘seconds’.

Ptolemy’s calculations also led him to an approximation for pi which he gave as 3.1416 in modern notation.

A contribution of Ptolemy to mathematics

A cyclic quadrilateral is a quadrilateral drawn with all four of its vertices on the circumference of a circle.

ABCD is a cyclic quadrilateral drawn inside the circle centre O.
ABCD is a cyclic quadrilateral drawn inside the circle centre O.

Ptolemy gave the following theorem named Ptolemy’s Theorem:

If ABCD is a cyclic quadrilateral, then the product of the lengths of the diagonals is equal to the sum of the products of pairs of opposite sides.

i.e. AC × BD = (AB × CD) + (AD × BC)

The proof of this theorem uses similar triangles.

It also requires knowledge of another well-known theorem variously called The Angles subtended by the same arc theorem or The Angles in the same segment theorem which states that:

If CD is an arc of a circle and A and B are any other two points on the circumference of the circle on the same side of chord CD, then Angle DAC = Angle DBC

Proof of Ptolemy’s Theorem

Draw in the diagonals AC and BD and find the point E on AC such that Angle ABE = Angle DBC

 

Equal angles are shown on the above diagram as follows:

Angle ABE =Angle DBC = Angle DAC (one arc)

Angle BAC = Angle BDC (two arcs)

Angle ACB = Angle ADB (three arcs)

Now consider two pairs of similar triangles:

Comments

    0 of 8192 characters used
    Post Comment

    No comments yet.

    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)