An Early Navigational Instrument
‘Rabbi Gamliel Used His Tube To Measure…’
The Gemara tells us that Rabbi Gamliel had an instrument which he used to measure the 2,000-amah Shabbos boundary (see Rashi). The Geonim (cited in Meiri; Teshuvos Hageonim 28:314) explain that his tool was a narrow tube. When looking directly downwards through a tube, one can only see one’s feet. As one raises the tube, one can see farther away. An object was placed at an exact distance of 2,000 amos from the holder of the tube. The tube was slowly raised until it could see the bottom of the object, but no farther. Rabbi Gamliel marked the angle that the tube was held at which the object was thus viewed. From then on, he was able to determine the distance of the 2,000-amah Shabbos boundary by holding the tube at the same angle. This method was effective only when viewing objects on a straight plane.
High School Trigonometry
In his commentary to the mishnah, the Rambam seems to suggest that the instrument was more complicated than this. He writes that there is no need to explain at length the specifics of this mechanism. Those familiar with trigonometry, the study of triangles and the proportions between their sides and angles, will understand how these principles are applied in navigation. Those unfamiliar with trigonometry will not fully understand how Rabbi Gamliel’s implement worked. There are six parts to any triangle: three sides and three angles. According to the calculations developed by mathematicians, in almost all cases, any three parts whose measures are known can be used to find the measures of the other three parts, if at least one of the known parts is a side. It was such a computation that Rabbi Gamliel used in determining the distance of techum Shabbos.
Based on the Talmud Yerushalmi (Eiruvin 4:2), some suggest a third possibility of how Rabbi Gamliel’s instrument worked. As we know, the closer an object is, the larger it appears. It therefore takes up a larger portion of our field of vision. A building on a distant skyline takes up only a small part of our view. When we stand face up to it, it blocks our sight entirely.
Measure Distances With Fingers
Rabbi Gamliel had a tube with a simple piece of glass at its end, which did not magnify his vision. On this glass, he made markings equally distant from each other. For the sake of explanation, let us say that they were one millimeter apart. When an object was distant, it would cover only one marking, a small part of his view. When it drew closer, it would cover many markings, thus a larger part of his view.
When using such an instrument, the length of the tube also determines how many markings would be covered by a distant object. To demonstrate, hold your hand adjacent to your face, with fingers separated, and look at a distant object through the crack between two fingers. Slowly, draw your hand away from your face and you will see that the same object will appear through all the cracks of your hand. Similarly, with a shorter tube, the glass is closer, the markings appear larger, and the object viewed at a distance appears to equal only one of them. With a longer tube, the glass is farther, the markings appear smaller, and the object viewed appears to equal many of them.
Start With The Right Measure
Rabbi Gamliel would use this tool to measure the boundary of the techum Shabbos. He would take a pole that he knew to be one meter tall, place it a distance of 1,000 meters, and note that it appeared to equal one millimeter on his glass. He could then calculate that the object in sight was at a distance of exactly 1,000 times the length of the tube. This simplified measuring the boundary. Rather than drawing strings, and counting their lengths, an object of known height could be sighted from afar, and its distance calculated based on how large it appeared in relation to the markings on the glass.
In previous generations, the Romans, in application of precisely this principle, would build towers of set heights next to their ports in order to help navigators on incoming ships measure their distance from shore.