TO FIND THE DISTANCES OF PLACES.*

The longitude and latitude of the given places are ascertained. The excess difference between each is multiplied into itself and the products which are called squares (for the square of a number is that number multiplied by itself) are set down and the two squares added together and their square root extracted. This root is then multiplied into 56⅔ Karoh which is the extent of a degree according to the moderns, or into 66⅔ which is the degree according to the ancients, and the product is the distance of the two places from each other. As long as a variation in the extent of longitudes and latitudes arises, the excess is multiplied proportionately and the result ascertained; where the longitudes and latitudes are equal, the rule does not hold good. This distance is calculated on the straight line, but some discrepancy will occur from the curve in direction. Abu Raiḥán Birúni has calculated this approximately and added a fifth of the result found.