A hundred years before, the Portuguese had come across an ingenious Arab device for telling the elevation of the sun. It consisted of a board with a knotted length of string run through the middle. If a mariner held the board vertically and sighted the horizon at one end and some object in the sky at the other, the length of the string between the board and his eye could be used to calculate the elevation of the object. In a short time a version appeared in Europe--with a second board replacing the string--called the cross-staff. However, since locating both the horizon and a star was almost impossible at any time except dawn or dusk, this device worked best for sighting the sun--save that it required staring into the disc of the sun to find its exact center. Also, the cross-staff could not be used when the sun was high in the sky, which was the case in equatorial waters. Another version of the cross-staff was the astrolabe, a round brass dial etched with degree markings and provided with a movable sight that permitted taking the elevation of the sun by its shadow. But even with the astrolabe there was the problem of catching the sun precisely at midday. And on a rolling ship the error in reading it could easily be four degrees. For longitude, a ship's location east or west on the globe, there were no fixed references at all; but as a mariner traveled east or west, the sun would come up somewhat earlier or later each day, and precisely how much earlier or later could be used to compute how far he had gone. Therefore, calculating longitude depended solely on keeping time extremely accurately--something completely impossible. The best timekeeping device available was the hourglass or "sandglass," invented somewhere in the western Mediterranean in the eleventh or twelfth century. Sandglass makers never achieved real accuracy or consistency, and careful mariners always used several at once, hoping to average out variations. But on a long voyage seamen soon totally lost track of absolute time. Since they were unable to determine a ship's location from the skies, mariners also tried to compute it from a vessel's speed and direction. Speed was estimated by throwing a log with a knotted rope attached overboard and timing the rate at which the knots in the rope played out--using a sandglass. Margins of error in computing speed were usually substantial. Direction, too, was never known completely accurately. A compass pointed to magnetic north, not true north, and the difference between these seemed to vary unaccountably at different locations on the globe. Some thought it had to do with the lodestone used to magnetize the needle, and others, like the Grand Pilot of the king of Spain, maintained seamen were merely lying to cover their own errors.