How to measure the size of the Earth Aristotle, the famous Greek natural philosopher, reports that mathematicians had allegedly evaluated the dimension of Earth at stadia, adding: `From their supposition, it follows that the shape of Earth must be a sphere and also that its size be small relative to the distance of other celestial bodies.‘ It was generally agreed upon that measuring the size of Earth could be done by measuring the altitude of a star from two cities situated on the same meridian.

Then, a difference expressed in degrees would be found. If the distance between the two cities was known, from estimates by caravaneers for instance, it would then be possible to find the value of a degree of meridian and hence derive the value of the terrestrial circumference. The stadium, Aristotle's unit length, apparently corresponds to 185 meters, so the value of km thus obtained is much too high. Archimedes, in his treatise on the number of sand grains quotes a value of stadia for the terrestrial circumference. This means that the measurement must have been attempted several times.

Eratosthenes' measurement From his readings, he had learnt that once a year (on the day of the Summer solstice), the bottom of a well situated at Aswan in Upper Egypt was illuminated by the Sun; However, at Alexandria, this never happened: obelisks always cast a shadow; He believed that Earth was a sphere;

He assumed that Alexandria and Aswan were on the same meridian; He knew (or better, he assumed) that the distance between the two cities was 5,000 stadia (as caravans covered the distance in 50 days at a rate of 100 stadia a day); He postulated that sunrays reached Earth as parallel beams (an idea that was commonly held by the mathematicians of his time).

Alexandria Aswan (Syene)

So, on solstice day, he decided to measure the length of the meridian shadow cast by a gnomon at Alexandria. He found a value of 1/50th of a circumference (i.e. 7.12°) and derived the value of the terrestrial circumference: 50 x = stadia. Although our idea of the exact value of the stadium (which was not the same at Athens, Alexandria or Rome) is fairly hazy, this puts the terrestrial circumference at km. The result is remarkable, although several errors were introduced in the calculations: The distance between Alexandria and Aswan is 729 km, not 800; The two cities are not on the same meridian (the difference in longitude is 3°); Aswan is not on the Tropic of Cancer (it is situated 55 km farther North); The angle difference is not 7.12° but 7.5°.

The most extraordinary thing is that the measurement rests on the estimated average speed of a caravan of camels: one can certainly do better in the matter of accuracy. Yet, in spite of all these flaws, it worked fine: around 250 BC, Earth had at last a size.

Stellar Parallax (Trigonometric Parallax ) Measuring distances to objects within our Galaxy is not always a straightforward task - we cannot simply stretch out a measuring tape between two objects and read off the distance. Instead, a number of techniques have been developed that enable us to measure distances to stars without needing to leave the Solar System. One such method is trigonometric parallax, which depends on the apparent motion of nearby stars compared to more distant stars, using observations made six months apart.

A nearby object viewed from two different positions will appear to move with respect to a more distant background. This change is called parallax. By measuring the amount of the shift of the object's position (relative to a fixed background, such as the very distant stars) with observations made from the ends of a known baseline, the distance to the object can be calculated. A conveniently long baseline for measuring the parallax of stars (stellar parallax) is the diameter of the Earth's orbit, where observations are made 6 months apart. The definition of the parallax angle may be determined from the diagram below:

The position of a foreground star is measured when the Earth is at position A. 6 months later, the Earth has moved around the Sun to position B - this provides a baseline of 2AU. Compared to the more distant 'fixed' stars, the foreground star has moved on the sky by an angle, 2p.

If the parallax angle, p, is measured in arc seconds (arcsec), then the distance to the star, d in parsecs (pc) is given by: The only star with a parallax greater than 1 arcsec as seen from the Earth is the Sun - all other known stars are at distances greater than 1 pc and parallax angles less than 1 arcsec. When measuring the parallax of a star, it is important to account for the star's proper motion, and the parallax of any of the 'fixed' stars used as references.

1. Why we can’t use parallax method to measure to distance for further stars? The further the distance, the smaller the parallax will be. So the results won’t be accurate any more. 2. What is the angle difference between Alexandria and Aswan when we measure the size of the Earth? (Tuff!!!) It’s 7.12° (or more accurately will be 7.5° 3. What does ‘AU’ stand for? It stands for Astronomical Unit.