gravitywave sources

The difficulty of measuring space and time

Notes

In antiquity, the positions of the stars could be measured, albeit crudely. Observations had shown that when one travelled north or south, the stars visible on any night of the year change. Further north, stars within a broad circle around the Pole Star are visible at all seasons, that is, they never set below the horizon. As you go south, this circle shrinks, but a greater number of equatorial stars become visible only sometimes and at some seasons. Contrastingly, when travelling east and west it was found that the same stars were visible at all locations along what could thus be defined as a line (really a circle) of latitude.

Similarly, measurements of the shadow cast by a gnonom – an upright stick in full sun – would give different results at different latitudes. By travelling up the River Nile past Aswan, one entered the tropical region, where the summer sun could be directly overhead. Travellers’ tales from the far northern seas suggested that in the summer there, the sun never set. Not everybody was willing to believe this sort of thing, but it was consistent with the spherical model described.

By these means the idea of latitude and longitude as seen in the stars – whose angular positions were already measured in these terms – could be projected onto the Earth’s surface. The simplest measure of latitude was the length of daylight at each particular season of the year. In addition, theories about climatic zones had been expounded from Aristotle to Marinos of Tyre, which Ptolemy drew on to deduce that localities sharing the same climate must be at approximately the same distance from the equator (B&J p.13). However he was less fixed in his ideas of these klimata (literally ‘inclination’ – used to refer to the climatic zones) than some earlier authors.

There was, unfortunately, no effective method for measuring longitude. Theoretically it was clear that as the sun went around its course it would reach the moment of noon simultaneously at some circle of points from pole to pole (called a ‘meridian’), and move on successively to illuminate other meridians from east to west. The time difference between noon sightings defined the angular difference in the east-west direction between any two places. Ptolemy (and others) knew this, but had very little to go on as far as direct measurement was concerned.

Without fast means of communication it was very hard to work out the time difference between two points. One of the isolated data-points available was a report of a total lunar eclipse seen by Alexander the Great’s army at Erbil, eleven days before their great victory over Darius III of Persia at Gaugamela in 330 BZ (Before Zero – in my opinion, it's high time we stopped messsing about with a silly system where there is no year zero, so my 'BC' dates might look off by one). The eclipse was recorded by Babylonian astronomers at the fifth hour of the night, and also at Carthage in north Africa at the second hour of the night. These two locations, being three hours apart in time difference, should therefore be 45° apart in longitude. Given that the true difference, as measured today, is about 34° we can what inaccuracy they dealt with.

This brings up the notion of measuring time. B&J address some aspects of time-measurement in their introduction, but not in great detail. It’s a subject I will no doubt come back to later. The water clocks and candle clocks of the period were both subject to great uncertainty. In addition, not everybody used hours of equal length – it was more usual to divide the night-time into twelve parts called hours, and the day-time similarly. Thus when something is said to happen at a certain hour, not only is this plus-or-minus thirty minutes even supposing equal hours, but the hour itself may be any length from the winter minimum to the summer maximum for the location concerned.

Measuring time is fundamental to the evidence base which Ptolemy had to work with. The longitude is effectively equivalent to the problem of measuring time accurately, which is well-known to have resisted full solution until the eighteenth century. Even for the latitude, which was in theory easier to observe, the commonest piece of information available was day-length. Most ordinary travellers would not be conducting a gnomon experiment, but they might try to estimate the number of hours of daylight (they would need to employ equal hours and not seasonal hours for this to make any sense). Day-length as a proxy for latitude is just as dependent on the measurement of time as the longitude.

Therefore for all its theoretical robustness, Ptolemy’s plan could only be partly achieved because of paucity of data. Unhelpfully, perhaps, he doesn’t face up to this lack of information but often gives a very precise-looking value for a location for which in truth he has little more to go on than a guess. The nineteenth-century commentator Edward Bunbury (History of Ancient Geography, 1879) is especially scathing about his general inaccuracy and especially these attempts at false precision. B&J are more understanding in their approach, because their concern is less focused on the actual map he was able to produce and more on the theoretical framework by which he could do so.

Having considered time and the ascertainment of longitude and latitude, we must consider distance units. The most important unit was the Greek stadion (usually called a ‘stade’ in English). Schiöth (2022) finds estimates among modern authors for the length of the stade ranging from 157 metres to 197 metres. However my two main sources for this investigation, Bunbury 1879 and B&J both settle on 185 metres.

B&J comment in a note:

There has been much disagreement concerning whether there was a single standard stade employed by all the geographical authors, and how large it was. We agree .. that they all used – or at least believed that they were using – the so-called Attic stade (p. 14).

When Ptolemy drew on other sources he treated the Roman mile (approx 1480 metres) as equal to 8 stades, and the Persian parasang or Egyptian schoinos as 30 stades. The parasang, a widely used unit then and later, was often regarded as an hour’s journey on foot and therefore varied with the terrain. It does not seem Ptolemy made allowance for this.

The final point is to make a connection between the longitude or latitude coordinates, and the measurement of travel distances. This relies on an estimate of the size of the Earth. If, for example, we know the circumference of the Earth, then we can calculate the size of a one degree difference of latitude or longitude. Thereby we can relate the distance between two places to their coordinates. If we know one, we can determine the other.

Ptolemy determined an estimate for the Earth’s circumference of 180,000 stades, equal to 33,300 km at 185 metres per stade. That’s about 17% less than modern measurements, and was also smaller than the more widely-accepted estimate in ancient times, that of Eratosthenes in the third century BZ, which was 252,000 stades. If we assume that Eratosthenes was using the same length for the stade then his estimate is about as far wrong on the big side as Ptolemy is on the short side. Some modern authors (e.g. Russo 2004) find it convenient to assume Eratosthenes used the smallest kind of stade, around 157 metres, thereby making his estimate almost exactly right!

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