ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Tartaglia

Updated on February 23, 2012

Tartaglia was one of the most colourful characters in the History of Mathematics.

Born Niccolò Fontana in the northern Italian town of Brescia in around 1499, he was given the name Tartaglia (which means ‘stammerer’ in Italian) following an incident in 1512 that left him with a lifetime imperfection of speech.

In 1512 Brescia was invaded by the French. Niccolò and other inhabitants of Brescia sought sanctuary in the cathedral, but the French soldiers pursued them and many of the people were massacred. Over 45,000 residents of Brescia were killed. Niccolò’s father, Michele Fontana, was one of those killed, and the boy was badly injured with a split skull and a sabre cut that passed through his jaw and palate. This resulted in the speech impediment that led to his name Tartaglia.

With the death of Michele, the family became impoverished. Tartaglia’s mother sent him to school for a few days, but could not afford to pay for regular schooling. So the boy stole a book, from which he taught himself to read and write.

That Tartaglia taught himself not only to read and write, but how to use and apply mathematics, is a mark of his genius.

He lectured at Verona before being appointed to the chair of mathematics at Venice in 1535.

Among Tartaglia’s most significant achievements were his translations into Italian of works of Archimedes and Euclid; in particular the1543 translation of Book V of Euclid’s Elements. Tartaglia used a Latin translation of the original Greek, rather than the commonly used Arabic translations that contained some inaccuracies, especially in relation to the Eudoxus’ Theory of proportions. Tartaglia’s commentary on this theory was later used by Galileo.

Tartaglia’s works include Nova Scienza (1537) in which he investigated the fall of bodies under gravity, Inventioni (1546) which contains among other things his solution of the cubic equation, an arithmetic treatise Trattato de numeri e misuri (1556), and a two volume treatise on arithmetic, published in 1560. In this he developed a method to obtain binomial coefficients using a triangle similar to Pascal's Triangle,

Tartaglia also applied mathematics to the science of artillery fire, and adapted Heron’s formula for the area of a triangle to give an expression for the volume of a tetrahedron given the lengths of its sides - Tartaglia’s Formula.

By Tartaglia’s time, the solution to the general quadratic equation ax2 + bx + c = 0 was well known. In modern notation we give the solutions in the form of the Quadratic Equation Formula:

Tartaglia’s most memorable achievement was his contribution to the solution of the general cubic equation ax3 + bx2 + cx + d = 0.

As stepping stones on the way to solving the general cubic equation, mathematicians tacked two simpler cases:

Case 1: x3 + mx = n where the coefficient of x2 is zero and

Case 2: x3 + px2 = n where the coefficient of x is zero.

Case 1 was successfully solved in 1515 by Scipione del Ferro, professor of mathematics at Bologna University. Del Ferro did not publish his result, but shared his discovery with his pupil Antonio Fior.

In 1535 Tartaglia announced that he had solved Case 2. However, Fior did not believe Tartaglia had a solution and challenged him to a public contest to solve cubic equations. Taking up the challenge, Tartaglia set himself the task of solving Case 1 also, which he did before the contest. Having solved both types, Tartaglia easily won the contest against Fior.

Later Tartaglia shared his solutions with Girolamo Cardano who promised not to publish. However, when Cardano learned of del Ferro’s earlier solution of Case 1, he decided to publish the solutions in his famous treatise Ars Magna, including Tartaglia’s solution of Case 2. This was followed by an acrimonious dispute between Tartaglia and Cardano.

Tartaglia died in Venice in 1557.

A contribution of Tartaglia to mathematics

In modern notation, Tartaglia gave the following solution to a cubic equation of the type

x3 + mx = n (Case 1 above):

Example Solve the cubic equation x3 + 9x = 6

In this case m = 9 and n = 6

Therefore

Of course a cubic equation always has three solutions, but in this case the other two solutions are complex numbers.

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)