pons asinorum

Pons asinorum is Latin for “bridge of asses”. During medieval times, this name was given to the fifth proposition in the first book of Euclid’s The Elements. In the original Greek, this proposition reads:

$T\tilde{\omega }\nu$ $\iota \sigma o\sigma \kappa \epsilon \lambda \tilde{\omega }\nu$ $\tau \rho \iota \gamma \acute{\omega }\nu \omega \nu$ $\alpha \iota$ $\pi \rho \stackrel{`}{o}\varsigma$ $\tau \tilde{\eta }$ $\beta \acute{\alpha }\sigma \epsilon \iota$ $\gamma \omega \nu \acute{\iota }\alpha \iota$ $\iota \sigma \alpha \iota$ $\alpha \lambda \lambda \acute{\eta }\lambda \alpha \iota \varsigma$ $\epsilon \iota \sigma \acute{\iota }\nu ,$ $\kappa \alpha \stackrel{`}{\iota }$ $\pi \rho o\sigma \epsilon \kappa \beta \lambda \eta \theta \epsilon \iota \sigma \tilde{\omega }\nu$ $\tau \tilde{\omega }\nu$ $\iota \sigma \omega \nu$ $\epsilon \upsilon \theta \epsilon \iota \tilde{\omega }\nu$ $\alpha \iota$ $\upsilon \pi \stackrel{`}{o}$ $\tau \stackrel{`}{\eta }\nu$ $\beta \acute{\alpha }\sigma \iota \nu$ $\gamma \omega \nu \acute{\iota }\alpha \iota$ $\iota \sigma \alpha \iota$ $\alpha \lambda \lambda \acute{\eta }\lambda \alpha \iota \varsigma$ $\epsilon \sigma o\nu \tau \alpha \iota .$

A translation of this proposition is:

In isosceles triangles, the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

There are a couple of reasons why this proposition was named pons asinorum:

• Euclid’s diagram for this proposition looks like a bridge.

• This is the first nontrivial proposition in The Elements and thus tests a student’s ability to understand more advanced concepts in Euclidean geometry. Therefore, this proposition serves as a bridge from from the trivial portion of Euclidean geometry to the nontrivial portion, and the people who cannot cross this bridge are considered to be unintelligent.

For more details, please see http://planetmath.org/?op=getmsg&id=15847a post written by rspuzio and http://planetmath.org/?op=getmsg&id=15849a post written by Wkbj79.