# How to measure the size of the Earth

This project invites you to measure the circumference of the Earth, in a collaboration with other Astronomy On-Line groups. To do so, you will have to read carefully the instructions given here and then to contact other groups which are interested in this type of project.

You may wish to contact actively those groups which are located more or less at the same geographical longitude as your own. But this is not an absolute condition.

You may also place a message about your interest in the Astronomy On-Line Communications Archive. You may do so via the Marketplace (Group Communications' Shop).

The measurement is not very difficult, and as long as the weather is not too bad and you can see the Sun, you should be able to obtain quite accurate results.

The organisers shall be happy to hear about your experience and look forward to your report(s). They will be brought in the Astronomy On-Line Newspaper.

Good luck!

## How big is the Earth?

Before Man started pondering over the question, it undoubtedly had been necessary to realize first that Earth was spherical. This can easily be understood during an eclipse of the Moon when one can see that the shadow cast by Earth on the Moon is a portion of a disk.

Aristotle, the famous Greek natural philosopher, reports that mathematicians had allegedly evaluated the dimension of Earth at 40.000 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 74.000 km thus obtained is much too high. Archimedes, in his treatise De Arenae Numero [On the number of sand grains] quotes a value of 300.000 stadia for the terrestrial circumference. This means that the measurement must have been attempted several times.

### Eratosthenes' measurement

Because he had been appointed Director to the Great Library at Alexandria by Ptolemaeus III Evergetes, Eratosthenes had an access to innumerable sources of knowledge.

He apparently made use of writings by Posidonius and reasoned thus:

• From his readings, he had learnt that once a year (on the day of the Summer solstice), the bottom of a well situated at Syene 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 Syene 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).
 Information on ERATOSTHENES Born at Cyrene 275 BC Studied at Alexandria and Athens Appointed Director of the Alexandrian Library 236 BC Got blind 195 BC Starved himself to death 194 BC

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. 7o 12') and derived the value of the terrestrial circumference: 50 x 5.000 = 250.000 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 40.000 km. The result is remarkable, although several errors were introduced in the calculations:

• The distance between Alexandria and Syene is 729 km, not 800;
• The two cities are not on the same meridian (the difference in longitude is 3o);
• Syene is not on the Tropic of Cancer (it is situated 55 km farther North);
• The angular difference is not 7o 12' but 7o 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.

Figure 1

Figure 2

### Picard's measurement

The idea of measuring Earth kept running in the minds of scientists but there was no improvement in the accuracy of the measurements until Galileo and the use of the telescope for astronomical purposes. A few years later, a team of the Royal Academy of Sciences in Paris decided to measure the value of the terrestrial radius. Picard, who had been assigned the task, was to measure as accurately as possible the linear distance between two points situated onthe same meridian and whose latitudes differed by 1o. Then the distance that had been measured would be multiplied by 360, thus yielding the value of the terrestrial circumference.

The limits of the arc to be measured were 6 km from LaFert-Alais, a small city North of Paris on one side and 20 km south of Amiens on the other side. The problem was to use a unit length that would be accepted by everybody. Picard's idea was quite clever: using the length of a pendulum oscillating seconds (mean solar time). Unfortunately, he did not know that the length of such a pendulum varies with latitude, which ruined all his efforts.

Anyhow, with a rigorous method and a concern for accuracy that remain exemplary, he set to work finally publishing in 1671 a treatise of about 30 pages, entitled Mesure de la Terre [The measurement of Earth]. The length of a degree of meridian was set at 57.057 toises, i.e. between 111 and 112 km, corresponding to a terrestrial radius of 6372 km.(1)

Earth had at last been measured with some more accuracy but there remained a lot to discover.

### A "revolutionary" Earth

The French Revolution burst out at the end of the XVIIIth century and it established a new system of scientific education. Important decisions were made regarding the units of measurements. To define the meter, the new universal standard, it was decided to measure a part of a terrestrial meridian. The adventurous task was led by Jean-Baptiste Delambre and Pierre Mechain, from 1792 to 1799, between Dunkirk (northern end) to Perpignan (southern end).

Similar missions had previously taken place:

• in Lapland, with Maupertuis, Clairaut, Camus and Lemonnier;
• in Peru, with Godin, Bouguer, la Condamine and one of the Jussieu brothers.

Much later, on September 3, 1957, the Toronto Colloquium of the International Association of Geodesy and Geophysics assessed the results of three centuries of measurements:

• semi-major axis of the reference ellipsoid: 6 378 245 m
• polar flattening: 1/298.3

Since then, the problem has been dealt with the help of satellites, and as the measurements became more and more accurate, it became more and more complex. But this is another story!

Determinations of the polar flattening through the centuries

 ` ` Newton: 1/230 Huygens: 1/578 1733 Cassini: -1/284 1737 Maupertuis: 1/178 1810 Delambre: 1/308 1841 Bessel: 1/299 1880 Clarke: 1/293 1924 Hayford: 1/297 1957 I.A.G. 1/298.257

Figure 3: During an eclipse of the Moon, it is possible to evaluate the ratio between the size of Earth's shadow and the size of the Moon (LUN207).

Figure 4: Seen from space, Earth seems to be an ideal sphere, but an improvement in the accuracy of the measurements has shown that this ideal shape was a mere appearance. The polar flattening can hardly be noticed on the image, but remote-sensing satellites can measure it (TER207).

Figure 5: Thanks to remote-sensing satellites as ERS1, a new vision of Earth is now available. They can monitor variations over more restricted zones as on this view showing the Atlantic Ocean (ATER0439).

The codes in the figure captions above correspond to references in the Geospace Picture Library (address below). These images have been digitized for inserting in a text file. Click on the images to see larger JPG-versions (but beware of the sizes of those files (14k, 144k, and 70k, respectively)!

#### Annotation:

(1) These values can be compared with present measurements, i.e.

 - mean equatorial radius: 6378 km - mean polar radius: 6357 km

## How two groups may measure the size of the Earth

Here is then how your group, together with another group, will be able to repeat this fundamental measurement.

• Requisites

2 vertical poles of the same height (think of sports equipment’s), one for each site. 1 telephone set or any computer allowing an Internet login.

• Method

In two separate sites distant of at least a few hundreds of kilometers and situated on the same meridian (say, for example: Lille and Montpellier in France), on any given day and at the same time, the shadows cast by the Sun are measured and the results are shared through a telephone or Internet link. If the two cites are on a different meridian, the mathematics involved will be a little more complicated.

• Calculation

As in the case of Eratosthene's experiment, there is very little math involved. = l - m Angle A = angle (l) measured on one site - angle (m) measured on the other site

Figure 6

A represents a part of the terrestrial circumference. The only thing to do is then to extrapolate, knowing the distance ML between the two sites.