Estimating the distance by Stick Method in a ۱۲th century Arabic manuscript

The stick method is a common and easy way to estimate the object size or its distance to an observer. Today, painters, scouts, artillery observers, woodchoppers, etc. use this method when they have no special rangefinder instrument. We don’t know the exact origin of this method but Euclid has described a similar way in proposition ۲۲ of his optics. Great astronomer of the ۱۲th century, cAbd al-Raḥmān al-Khāzinī, has explained the stick method in his treatise named about the marvelous instruments (fī Ālāt al-cAjība). He dedicated his treatise to explain observatory instruments and amplified how to use these instruments and the information gathered by them. al-Khāzinī described ۷ instruments used for observing the stars in ۷ independent books and added their application as the surveying tools. He presented the geometrical arguments to show us why and how we can use his formulas to gain mathematical information about subjects from data gathered by these instruments. al-Khāzinī talked about the stick method in an appendix without the geometrical arguments. His main goal was to describe a simple way for soldiers and horsemen to calculate their distance from their enemies or estimate the strength of foes armies. Furthermore, he explained estimating the distance from an object by knowing its diameter or vice versa and finally estimating the distance and diameter of the object when they were both unknown. His method was easy and practical for anyone in the military campaign who didn’t know complicated mathematical relations. He used a wooden stick about ۸۰ c.m long to do this job. In this paper, I briefly presented al-Khāzinī’s original treatise and four manuscripts available in the libraries and talked about how amended al-Khāzinī’s treatise in the first step. Secondly, I introduced the English translation of al-Khāzinī’s appendix about the stick method with its main Arabic text. Finally, I described his method by modern mathematics symbols and notations and then I tried to rebuild the geometrical arguments according to his previous book about the triquetrum.

Keywords:

About the Marvelous Instruments treatise , Euclid’s optics , estimating the distance , geometry , al-Khāzinī , stick method