CONTENTS
1.1 Geometry
1.2 Trigonometry
1.3 Calculus
1.3.1 Functions
1.3.2 Infinity in Maths
1.3.3 Derivatives, Integrals and the Fundamental Theorem of Calculus
1.3.4 Inverse Trigonometric Functions
1.4 Eratosthenes’s final calculation
1.5 Aristarchus’s calculations
THERE are several elementary concepts that need to be properly reviewed in order to fully understand the phenomena happening in our solar system. Obviously, we do not have here the space or the time to review the entirety of geometry, trigonometry, calculus, etc. However, especially for those readers that need to refresh concepts that haven’t been used in some time, we treat in some detail most of the useful mathematical concepts that will be used in later chapters, especially the chapter on celestial mechanics. Our treatment is by no means rigorous and is mostly used to prove statements about our world and, more specifically, our solar system. We will see in the next few sections how the knowledge of very simple geometry has allowed ancient Greek philosophers to state that not only the Earth is spherical, but also estimate the distance from the Earth to the Moon and the Earth to the Sun.
We will use the Greek philosophers’ amazing achievements as an excuse to introduce some of the concepts that are needed in the rest of the book putting considerable attention to show how maths is deeply used – we would say embedded – in the physical world. We will also often digress to show how systems and concepts apparently disconnected from the study of the solar system are actually used, sometimes so automatically that we forget how important they are. An example is the case of digital electronics and logic circuits inside computers.
Now imagine you are a Greek philosopher, named Eratosthenes (see fig. 1.1), residing in a beautiful city in Egypt (Alexandria). You are taking care of one of the most complete and important libraries in existence and you have time to read, study and think. Somebody has just told you that during one of his trips to Syene (today’s Aswan) in the south of Egypt, he has noticed something weird: at midday of the special day of summer solstice, i.e. the longest day and shortest night of the year, the Sun shone directly down a deep vertical well. In other words, looking directly down the well, your head blocks exactly the reflection of the Sun by the water at the bottom of the well. The majority of people when confronted with the news would simply think “that’s strange” and then go back to their lives. But you have time to think … and you run outside at midday of the summer solstice in your city of Alexandria and plant a vertical stick in the ground to check if the Sun is overhead in Alexandria as well. It is not! You have time to think … and all of a sudden you know what all means: the Earth is not flat, it is a sphere! Not only that, but you are capable...
