ARISTARCHO'S PROPORTIONSRosa M. Ros, Eder Viñuales

The proposal of this activity is to repeat both experiments with students. The idea is to repeat the mathematical process designed by Aristarchos and Eratostenes and, if possible to repeat the observations.
We know that he made a mistake, possibly because it was very difficult for him to determine the instant of the quarter phase. In fact this angle is 89º 51', but the process used by Aristarchos is correct. In figure 1, if we use the definition of the sine, it can be deduced that sin 9' = ES/EM, where ES is the distance from the Earth to the Sun, and EM is the distance from the Earth to the Moon. Then approximately,
Relationship between the Earth's distances, the Moon's radius, the Sun's radius and the radius of the Earth
During a Moon's eclipse, Aristarchos observed that the time necessary in order for the Moon to cross the cone's shadow of the Earth was double the time necessary for the Moon's surface to be covered.
Therefore he deduced that the shadow of the Earth's diameter was double the Moon's diameter, that is to say, the relation of both diameters or radii was 2:1. Really we know that this value is 2.6:1.
Then we deduce the following relationship
where x is an auxiliary variable.
Introducing in this previous expression the relationships ES = 400 EM and R_{S}=400 R_{M}, it is possible to eliminate x and simplifying it,
R_{M} = R_{E} 401/1440
is obtained, which offer the opportunity to express all the dimensions concerning the Earth's radius, so
Secondly, during an eclipse of the Moon, using a chronometer, it is possible to calculate the relation of times between: "the first and the last contact of the Moon with the Earth's shadow", that is to say, measuring the diameter of the cone's shadow of the Earth (Fig. 3) and "the time necessary to cover the Moon surface", that is to say the measure the Moon's diameter.
Finally it is possible verify if the relationship between both times is 2:1 or 2.6:1
The most important objective of this activity is not the result obtained. The most important thing is that it suggests to the students that if they use their knowledge and intelligence, they can obtain interesting results with reduced facilities. In this case the ingenuity of Aristarchos was important in obtaining some concepts of the size of the EarthMoonSun system.
It is a good idea to measure the Earth's radius with students according to the process used by Eratostens. Although Eratostenes' experiment is very wellknown, we present here a short version in order to complete the previous Aristarchos experiment.
We consider two sticks introduced perpendicularly in the soil, in two different cities of the Earth's surface on the same meridian. The sticks must be pointed towards the Earth's centre. Normally it is better to use a plumb where we mark a point to measure longitudes. We need to measure the longitude on the plumb from the soil to this mark, and the longitude on its shadow from the base of the plumb to the mark's shadow.
We can consider that the solar rays are parallel. Those Sun's rays produce two shadows for each plumb. We measure the longitudes of the plumb and its shadow. Using the definition of tangent we can obtain the angles a and b .
The central angle g can be calculated because the sum of the three angles of a triangle is equal to p radians. Then :
where a and b can be obtained from measuring the plumb and its shadow.
Finally by proportionality between the angle g and the longitude of its arc d (determined by the distance on the meridian between the two cities), and 2p radians of the meridian circle and its longitude 2 p R_{E}, that is to say g/d = 360/(2 p R_{E}), then
it can be deduced, where g is obtained from the observation and d is the distance in km between both cities. We can find d from a good map (for example the army maps are an excellent source for obtaining it).
Also the objective of this activity is not to obtain a precise result. We would only like the students to discover that by thinking and using all the resources that they can imagine, they can obtain surprising results.